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物理代写|粒子物理代写Particle Physics代考|C P T symmetry

The Dirac equation is by construction invariant under space inversions, or parity transformations $(\mathcal{P})$. We just showed the existence of a second symmetry transformation, that of charge conjugation $(\mathcal{C})$. We wish to investigate here the consequences of time reversal $\mathcal{T}$, with $(\mathcal{T} x)^\mu=(-t, \boldsymbol{x})$. We can easily check that $\gamma^0 \gamma^5 \Psi^C(\mathcal{T} x)$, which is an anti-unitary operation, also satisfies the Dirac equation. It is interesting to consider the product of these three symmetries, namely the product of ${ }^5$
$$\begin{gathered} \mathcal{P}: \Psi(x) \rightarrow \mathrm{i} \gamma^0 \Psi(\mathcal{P} x) \ \mathcal{C}: \Psi(x) \rightarrow \Psi^C(x)=C \gamma^0 \Psi^(x) \end{gathered}$$ and $$\mathcal{T}: \Psi(x) \rightarrow \gamma^0 \gamma^5 \Psi^C(\mathcal{T} x)$$ where $(\mathcal{P} x)^\mu=(t,-\boldsymbol{x})$ denotes space inversion. We can verify that the product $C P T$ of the three operators taken in any order is an invariance of the theory. Indeed, start with \begin{aligned} P C T \Psi(x) & =\mathrm{i} \gamma^0\left(C \gamma^0\left[\gamma^0 \gamma^5 C \gamma^0 \Psi^(\mathcal{P} \mathcal{T} x)\right]^*\right) \ & =\mathrm{i} \gamma^0\left(C \gamma^0\left(\gamma^0 \gamma^5 C^{-1} \gamma^0 \Psi(-x)\right)\right) \ & =\mathrm{i} \gamma^5 \Psi(-x) \end{aligned}
and ask the following question: Given a wave function $\Psi(x)$, which satisfies the Dirac equation in an external electromagnetic field $A_\mu(x)$, can we find the equation satisfied by its $C P T$-transformed wave function $\mathrm{i} \gamma^5 \Psi(-x)$ ? The answer is simple:
\begin{aligned} 0 & =\left[\mathrm{i} \not \partial_x-e \not A(x)-m\right] \Psi(x) \ & =\mathrm{i} \gamma^5\left[\mathrm{i} \not \partial_x-e \not A(x)-m\right] \Psi(x) \ & =\left[-\mathrm{i} \not \partial_x+e \not A(x)-m\right] \mathrm{i} \gamma^5 \Psi(x) \ & =\left[\mathrm{i} \not \partial_x+e \not A(-x)-m\right] C P T \Psi(x) \end{aligned}
where, in the last step, we have performed the change of variable $x^\mu \rightarrow-x^\mu$.

物理代写|粒子物理代写Particle Physics代考|Chirality

In section 7.3.6 we introduced the so-called “standard” representation, which separates the components of a Dirac spinor that vanish in the $c \rightarrow \infty$ non-relativistic limit. At the other end, at the ultra-relativistic limit where the mass is negligible, we expect, at least in the absence of any external potential, the Dirac equation to split into a pair of Weyl equations. We shall introduce a formal way to demonstrate this fact.

We define two orthogonal projectors $P_L$ and $P_R, L$ and $R$ standing for “left” and “right” respectively, by
$$P_L=\frac{1+\gamma^5}{2}, \quad P_R=\frac{1-\gamma^5}{2}$$
They are Hermitian operators and satisfy the projection, orthogonality and completeness relations: $P_{L, R}^2=P_{L, R}, P_L P_R=P_R P_L=0$ and $P_L+P_R=1$. With the help of these projectors we define the “left” and “right” components of a general Dirac spinor $\Psi(x)$ by
$$\Psi_{L, R}(x)=P_{L, R} \Psi(x), \quad \Psi(x)=\Psi_L(x)+\Psi_R(x)$$
In the Weyl basis we used in equation (7.38), in which $\gamma^5$ is diagonal, $P_L$ and $P_R$ project onto the $\xi$ and $\eta$ spinors, respectively. Using $\Psi_L$ and $\Psi_R$, the free Dirac action (7.61) becomes
$$S=\int \mathrm{d}^4 x\left[\bar{\Psi}_L \mathrm{i} \not \partial \Psi_L+\bar{\Psi}_R \mathrm{i} \not \partial \Psi_R-m\left(\bar{\Psi}_L \Psi_R+\bar{\Psi}_R \Psi_L\right)\right]$$

物理代写|粒子物理代写Particle Physics代考|C P T symmetry

$$\left.\mathcal{P}: \Psi(x) \rightarrow \mathrm{i} \gamma^0 \Psi(\mathcal{P} x) \mathcal{C}: \Psi(x) \rightarrow \Psi^C(x)=C \gamma^0 \Psi^{(} x\right)$$

$$\mathcal{T}: \Psi(x) \rightarrow \gamma^0 \gamma^5 \Psi^C(\mathcal{T} x)$$

$$\left.P C T \Psi(x)=\mathrm{i} \gamma^0\left(C \gamma^0\left[\gamma^0 \gamma^5 C \gamma^0 \Psi^{(\mathcal{P} T} x\right)\right]^*\right) \quad=\mathrm{i} \gamma^0\left(C \gamma^0\left(\gamma^0 \gamma^5 C^{-1} \gamma^0 \Psi(-x)\right)\right)=\mathrm{i} \gamma^5 \Psi(-x)$$

$$0=\left[\mathrm{i}, \partial_x-e A(x)-m\right] \Psi(x) \quad=\mathrm{i} \gamma^5\left[\mathrm{i} \partial \partial_x-e A(x)-m\right] \Psi(x)=\left[-\mathrm{i} \partial \partial_x+e A(x)-m\right] \mathrm{i} \gamma^5 \Psi(x) \quad=\left[\mathrm{i}, \partial_x+e A(-x)-m\right] C P T \Psi(x)$$

物理代写|粒子物理代写Particle Physics代考|Chirality

$$P_L=\frac{1+\gamma^5}{2}, \quad P_R=\frac{1-\gamma^5}{2}$$

$$\Psi_{L, R}(x)=P_{L, R} \Psi(x), \quad \Psi(x)=\Psi_L(x)+\Psi_R(x)$$

$$S=\int \mathrm{d}^4 x\left[\bar{\Psi}_L \mathrm{i} \not \partial \Psi_L+\bar{\Psi}_R \mathrm{i} \not \partial \Psi_R-m\left(\bar{\Psi}_L \Psi_R+\bar{\Psi}_R \Psi_L\right)\right]$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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物理代写|粒子物理代写Particle Physics代考|The Klein-Gordon Equation

We start with the simplest Lorentz covariant wave equation, namely the Klein-Gordon equation. From our study of the Lorentz group, we expect it a priori to be relevant to the description of the quantum mechanics of a spinless particle. Since the wave function in quantum mechanics is complex valued, it is natural to start with the Klein-Gordon equation (7.1) for a complex function $\Phi(x)$
$$\left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\Delta+m^2\right) \Phi(t, \boldsymbol{x})=0$$
It is straightforward to verify that in the non relativistic limit, $c \rightarrow+\infty$, the KleinGordon equation reduces to the Schrödinger equation. To show this it is enough to extract the mass from the energy dependence of $\Phi(x)$. This is obtained by parametrising $\Phi$ in the following way:
$$\Phi(t, \boldsymbol{x})=\exp \left(-\frac{\mathrm{i} m c^2}{\hbar} t\right) \Psi(t, \boldsymbol{x})$$

Then, the Klein-Gordon equation becomes
$$\left(\frac{1}{c^2}\left(\frac{\partial}{\partial t}\right)^2-\frac{\mathrm{i}}{\hbar} 2 m \frac{\partial}{\partial t}-\Delta\right) \Psi(x)=0 .$$
In the non-relativistic approximation, $c \rightarrow+\infty$, the first term can be neglected, and we recover the free Schrödinger equation
$$\mathrm{i} \hbar \frac{\partial}{\partial t} \Psi(x)=-\frac{\hbar^2}{2 m} \Delta \Psi(x)$$
So, at first sight, the Klein-Gordon equation seems to have the right properties to give the relativistic generalisation of the Schrödinger equation. We suspect though that this could not be right because, if it were that simple, Schrödinger would have written directly this more general relativistic equation. Let us show that, indeed, interpreting $\Phi(x)$ as the particle wave function, does lead to physical inconsistencies.

物理代写|粒子物理代写Particle Physics代考|The Dirac Equation

Dirac derived his equation in 1928 as a relativistic generalisation of Schrödinger’s equation for the electron. Schrödinger’s equation is first order in time derivatives, but second order in derivatives with respect to the spatial coordinates. It does have a conserved probability current with positive definite density. We just saw that this property is lost in the obvious relativistic extension, the Klein-Gordon equation, which has second order derivatives with respect to both time and space. It is this difference, which motivated Dirac to look for an equation with first order derivatives. For a single scalar function there is no such non-trivial Lorentz covariant first order differential equation, so Dirac assumed a multi-component wave function and looked for a matrix equation. In doing so, he discovered the spinorial representations of the Lorentz group. In the notation we used in section 7.3 the equation reads
$$(\mathrm{i} \not \partial-m) \Psi(x)=0$$
As we have shown already, this equation is Lorentz covariant, provided the $\gamma$ matrices satisfy the Clifford algebra relation (7.41). It is obtained from the Lagrangian density given in equation (7.60).

物理代写|粒子物理代写Particle Physics代考|The Klein-Gordon Equation

$$\left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\Delta+m^2\right) \Phi(t, \boldsymbol{x})=0$$

$$\Phi(t, \boldsymbol{x})=\exp \left(-\frac{\mathrm{i} m c^2}{\hbar} t\right) \Psi(t, \boldsymbol{x})$$

$$\left(\frac{1}{c^2}\left(\frac{\partial}{\partial t}\right)^2-\frac{\mathrm{i}}{\hbar} 2 m \frac{\partial}{\partial t}-\Delta\right) \Psi(x)=0 .$$

$$\mathrm{i} \hbar \frac{\partial}{\partial t} \Psi(x)=-\frac{\hbar^2}{2 m} \Delta \Psi(x)$$

物理代写|粒子物理代写Particle Physics代考|The Dirac Equation

$$(\mathrm{i} \partial \partial-m) \Psi(x)=0$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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物理代写|粒子物理代写Particle Physics代考|Lagrangian, Hamiltonian and Green functions

The Lagrangian formulation of the Dirac equation presents some subtleties, because the latter is a first order differential equation. Recall the results we obtained for the case of a complex scalar field. The Lagrangian density (7.27) depends on the fields $\phi$ and $\phi^*$ as well as their first derivatives. This is consistent with the fact that the equations of motion are second order differential equations and we must assign initial values for the fields and their first derivatives. When we vary with respect to the field $\phi$ in order to obtain the Euler-Lagrange equations, we must also take into account the variation of $\partial \phi$. For the Dirac equation, however, which is a first order equation, we can vary only with respect to $\psi$ and $\bar{\psi}$, not to their derivatives. This implies in turn that the standard way to obtain the Hamiltonian through a Legendre transformation should be reformulated. In this section we want to present the rules which will make it possible for us to use the Lagrangian and Hamiltonian formalisms, without attempting a mathematically rigorous justification.

We choose the Lagrangian density corresponding to the Dirac equation in the form
$$\mathcal{L}{\mathrm{D}}=\frac{\mathrm{i}}{2}\left(\bar{\psi} \gamma^\mu \partial\mu \psi-\partial_\mu \bar{\psi} \gamma^\mu \psi\right)-m \bar{\psi} \psi$$
This is justified by the fact that we obtain the Dirac equations for $\psi$ and $\bar{\psi}$ as a consequence of the stationarity requirement of the action $S=\int \mathcal{L}{\mathrm{D}} d^4 x$ under independent variations of $\bar{\psi}$ and $\psi$, respectively. Because of the linear dependence of $\mathcal{L}{\mathrm{D}}$ on $\psi$ or $\bar{\psi}$, the action has neither a minimum nor a maximum. Thus, the overall sign of the action can be chosen at will. Provided that the field vanishes at infinity, we can rewrite the action as
$$S=\int \mathrm{d}^4 x(\mathrm{i} \bar{\psi} \not \partial \psi-m \bar{\psi} \psi)$$

物理代写|粒子物理代写Particle Physics代考|The plane wave solutions

We have shown that the solutions of the Dirac equation are solutions of the KleinGordon equation $\left(\square+m^2\right) \psi=0$ as well. Consequently, a plane wave solution $\psi(x) \sim$ $\exp (-\mathrm{i} k \cdot x)$ of the Dirac equation has to satisfy the condition $k^2=k_0^2-\boldsymbol{k}^2=m^2$, which means that its energy $k_0$ can have either sign. We shall be interested in the full set of plane wave solutions of both positive and negative energies, since only their union forms a basis. We fix the zero component of the wave vector $k^\mu$ to $k_0=+\sqrt{\boldsymbol{k}^2+m^2} \equiv E_k$. Then, we denote the positive energy solution of wave vector $\boldsymbol{k}$ by
$$\psi^{(+)}(x)=\mathrm{e}^{-\mathrm{i} k \cdot x} u(\boldsymbol{k})$$
and the negative energy one by
$$\psi^{(-)}(x)=\mathrm{e}^{\mathrm{i} k \cdot x} v(\boldsymbol{k})$$
where $u$ and $v$ are four-component spinors, whose components are labelled $u_r$ and $v_r$, $r=\mathbf{1}, \ldots, 4$
From $\left(\mathrm{i} \gamma^\mu \partial_\mu-m\right) \psi^{( \pm)}(x)=0$, we obtain
$$(\not k-m) u(\boldsymbol{k})=0 \text { and }(\not k+m) v(\boldsymbol{k})=0$$
Let us choose the $\gamma$ matrices in the standard representation. For $k=0$, the equations (7.72) simplify to
$$\left(\gamma^0-1\right) u(\mathbf{0})=0 \quad \text { and }\left(\gamma^0+1\right) v(\mathbf{0})=0$$
and lead to $u_3=u_4=v_1=v_2=0$. A possible basis of the solutions is
$$\hat{u}^{(1)}(m, \mathbf{0})=\left(\begin{array}{l} 1 \ 0 \ 0 \ 0 \end{array}\right), \quad \hat{u}^{(2)}(m, \mathbf{0})=\left(\begin{array}{l} 0 \ 1 \ 0 \ 0 \end{array}\right), \quad \hat{v}^{(1)}(m, \mathbf{0})=\left(\begin{array}{l} 0 \ 0 \ 1 \ 0 \end{array}\right), \quad \hat{v}^{(2)}(m, \mathbf{0})=\left(\begin{array}{l} 0 \ 0 \ 0 \ 1 \end{array}\right)$$

物理代写|粒子物理代写Particle Physics代考|Lagrangian, Hamiltonian and Green functions

$$\mathcal{L D}=\frac{\mathrm{i}}{2}\left(\bar{\psi} \gamma^\mu \partial \mu \psi-\partial_\mu \bar{\psi} \gamma^\mu \psi\right)-m \bar{\psi} \psi$$

$$S=\int \mathrm{d}^4 x(\mathrm{i} \bar{\psi} \partial \partial-m \bar{\psi} \psi)$$

物理代写|粒子物理代写Particle Physics代考|The plane wave solutions

$$\psi^{(+)}(x)=\mathrm{e}^{-\mathrm{i} k \cdot x} u(\boldsymbol{k})$$

$$\psi^{(-)}(x)=\mathrm{e}^{\mathrm{i} k \cdot x} v(\boldsymbol{k})$$

$$(k-m) u(\boldsymbol{k})=0 \text { and }(k+m) v(\boldsymbol{k})=0$$

$$\left(\gamma^0-1\right) u(\mathbf{0})=0 \quad \text { and }\left(\gamma^0+1\right) v(\mathbf{0})=0$$

$$\hat{u}^{(1)}(m, \mathbf{0})=\left(\begin{array}{l} 1 \ 0 \ 0 \ 0 \end{array}\right), \quad \hat{u}^{(2)}(m, \mathbf{0})=\left(\begin{array}{l} 0 \ 1 \ 0 \ 0 \end{array}\right), \quad \hat{v}^{(1)}(m, \mathbf{0})=\left(\begin{array}{l} 0 \ 0 \ 1 \ 0 \end{array}\right), \quad \hat{v}^{(2)}(m, \mathbf{0})=\left(\begin{array}{l} 0 \ 0 \ 0 \ 1 \end{array}\right)$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Particle Physics, 物理代写, 粒子物理

avatest™帮您通过考试

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物理代写|粒子物理代写Particle Physics代考|The Klein-Gordon Equation

We start with the simplest case, the equation for a real, scalar field. In our notation of Chapter 5 , it belongs to the trivial one-dimensional $(0,0)$ representation of the Lorentz algebra. In this case the elements which are at our disposal are the field itself $\phi$ and the four-vector operator of derivation $\partial_\mu$. It is clear that the lowest order, non-trivial, relativistically covariant equation, which can be built with these quantities is
$$\left(\partial_\mu \partial^\mu+m^2\right) \phi(x)=0$$
This is the Klein-Gordon equation. In our usual system of units $\hbar=c=1$ the parameter $m^2$ has the dimensions of $[\mathrm{M}]^2$. We shall often call this equation the massive Klein-Gordon equation, although at this stage we have no real justification for this name. The $m^2$ is just a parameter which can take any real value. ${ }^1$ This equation can be derived by the variational principle applied to the action
$$S[\phi]=\int \mathrm{d}^4 x \mathcal{L}(x)=\frac{1}{2} \int \mathrm{d}^4 x\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$
where
$$\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$
is the Lagrangian density. The canonical momentum associated to $\phi(x)$ is given by
$$\pi(x)=\frac{\partial \mathcal{L}}{\partial\left(\partial_0 \phi(x)\right)}=\partial_0 \phi(x)$$

and the Hamiltonian density by
$$\mathcal{H}=\frac{1}{2}\left[\pi^2(x)+\left(\partial_i \phi(x)\right)^2+m^2 \phi^2(x)\right]$$
Equation (7.1) admits plane wave solutions

物理代写|粒子物理代写Particle Physics代考|The Green’s functions

The solution of a linear homogeneous wave equation is always rather trivial. However, in practice we are often interested in the dynamics of the field $\phi(x)$ coupled to a given external source described by a function $j(x)$. The corresponding equation of motion is
$$\left(\square+m^2\right) \phi(x)=j(x)$$
It is an equation of hyperbolic type. As a second order differential equation, its solutions are determined by the Cauchy data, the value of the function and its first derivatives on a surface, called the Cauchy surface. In practice, every time we use local coordinates, we will take as a Cauchy surface the hyperplane $\mathbb{R}^3=\left{(t, \boldsymbol{x}) \in M^4 \mid t=0\right}$ or its time translations. By relativistic invariance, the properties of the solutions are independent of this particular choice and we can choose any space-like hypersurface.
Since the equation (7.10) is linear, the solution for a general $j(x)$ will be given by the superposition principle starting from the solution of the equation corresponding to a point source
$$\left(\square+m^2\right) G(x, y)=\delta^4(x-y)$$
In physics, these solutions are the so-called elementary solutions or Green functions. $G(x, y)$ is the field produced by a point source, which appears at the point $\boldsymbol{y}$ instantaneously at time $y_0$. We are particularly interested in those solutions that are translationally invariant. The general solution of $(7.10)$ will then be of the form
$$\phi(x)=\phi_0(x)+\int \mathrm{d}^4 y G(x-y) j(y)$$
where $\phi_0(x)$ is a solution of the homogeneous equation $\left(\square+m^2\right) \phi_0(x)=0$, which is fixed by the Cauchy data.

物理代写粒子物理代写Particle Physics代考|The Klein-Gordon Equation

$$\left(\partial_\mu \partial^\mu+m^2\right) \phi(x)=0$$

$$S[\phi]=\int \mathrm{d}^4 x \mathcal{L}(x)=\frac{1}{2} \int \mathrm{d}^4 x\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$

$$\mathcal{L}=\frac{1}{2}\left(\partial_\mu \phi(x) \partial^\mu \phi(x)-m^2 \phi^2(x)\right)$$

$$\pi(x)=\frac{\partial \mathcal{L}}{\partial\left(\partial_0 \phi(x)\right)}=\partial_0 \phi(x)$$

$$\mathcal{H}=\frac{1}{2}\left[\pi^2(x)+\left(\partial_i \phi(x)\right)^2+m^2 \phi^2(x)\right]$$

物理代写|粒子物理代写Particle Physics代晏|The Green’s functions

$$\left(\square+m^2\right) \phi(x)=j(x)$$

$$\left(\square+m^2\right) G(x, y)=\delta^4(x-y)$$

$$\phi(x)=\phi_0(x)+\int \mathrm{d}^4 y G(x-y) j(y)$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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物理代写|粒子物理代写Particle Physics代考|New Elementary Particles and New Quantum Numbers

As mentioned earlier, the charged $\pi$-meson, discovered in 1947 , was first seen in cosmic rays. ${ }^{47}$ Since the early 1950 s accelerators took over. They provided high incident fluxes under controlled energy and momentum conditions. Their extensive use in higher and higher energies yielded a rich harvest of new “elementary” particles. They were all unstable with lifetimes ranging from $2 \times 10^{-6} \mathrm{~s}$ (the $\mu$-lepton) to $\sim 10^{-23} \mathrm{~s}$ for the very short-lived hadrons. Obviously, these last ones cannot leave a visible track in the detector and were discovered as resonances, in the way we indicated in Chapter 4. At present the tables published by the Particle Data Group (PDG) contain several hundred entries.

In Chapters 2 and 4 we pointed out that the mean lifetime of an unstable state is the inverse of the width $\Gamma$ of the state, the latter being equal to its decay probability per unit time. The enormously wide span of the observed lifetimes indicates that all these decays are not due to one and the same underlying interaction. Indeed, we shall see in this book that we can distinguish roughly three classes of unstable particles according to the order of magnitude of their lifetimes.

• Particles with “long” lifetimes, longer than roughly $10^{-10} \mathrm{~s}$. We saw that the muons and the charged pions belong to this class. This means that they have a relatively small decay probability and the interactions responsible are the ones we called “weak” in section 6.4.3. An immediate consequence of this weak probability is that these particles live long enough to leave a measurable trace in the detector. A useful quantity is the product $c \tau$, where $c$ is the speed of light in the vacuum and $\tau$ the lifetime in the particle rest frame. For the muon it is almost $660 \mathrm{~m}$ and for the charged pion $7.8 \mathrm{~m}$. It follows that we can build beams out of these particles and use them as projectiles for scattering experiments.
• Particles with intermediate lifetimes of order $10^{-15} \mathrm{~s}$. These decays are due to the electromagnetic interactions. The neutral pion $\left(\tau \simeq 10^{-16} \mathrm{~s}\right)$ belongs to this class. The corresponding $c \tau$ value is on the order of $25 \mathrm{~nm}$, much too short to build a beam of $\pi^0 \mathrm{~s}$.
• Very short-lived particles with lifetimes shorter than $10^{-20} \mathrm{~s}$. The interactions responsible for their decays are the strong interactions. They leave no measurable track in the detector and they are observed as resonances. In fact, if we look at the tables, we see that it is more convenient for these particles to talk about their width $\Gamma$, rather than their lifetime $\tau .^{48}$

物理代写|粒子物理代写Particle Physics代考|Resonances

In Chapter 4 we introduced the concept of resonance in scattering as a sharp increase of the cross section around a certain value of the energy. This should be understood as an increase with respect to a certain background which is the expected cross section in the absence of such a resonance. In Chapter 12 we shall develop precise methods to estimate this background, but in most cases it is given by the phase space integral we introduced in equation (4.7). The first hadronic resonance was discovered in the late $1940 \mathrm{~s}$, but it took some time before the correct interpretation was given. It is a pion-nucleon resonance of mass $1232 \mathrm{MeV}$ and width around $115 \mathrm{MeV}$. It has spin and isospin equal to $3 / 2$, positive parity and in the particle tables appears today as $\Delta(1232) 3 / 2^{+}$. It was first seen as a sharp increase in the pion-nucleon cross section. It was soon followed by several other resonan ces of the pion-nucleon system, a fact which greatly confused people in their efforts to build a simple dynamical model for the interaction of pions and nucleons. In Figure 6.11, we show a 1961 plot with the evidence of resonances made out of three pions. The statistics is still poor and the errors quite large, but the resonance phenomena are clear. For comparison, we show a very precise resonance curve of the neutral weak gauge boson $Z$, which was obtained at LEP (Figure 6.12). Finally, Figure $22.11$ shows the discovery of the last elementary particle, the Brout-Englert-Higgs boson, at the LHC. In this case the background is not simply phase space but the result of a very elaborate calculation of the expected two photon events as a function of their invariant mass. As we see, the signal-tobackground ratio is very small, but it was compensated by the very accurate resolution in the invariant mass measurements combined with a very precise determination of the expected background. The excess of a few hundred events consistent with the decay of a resonance can be clearly seen.

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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物理代写|粒子物理代写Particle Physics代考|Heisenberg and the Symmetries of Nuclear Forces

As we have noted already, 1932, the year of the discovery of the neutron, is the year when nuclear physics started taking its present form. Of course, the transition from the old nuclear model of protons and electrons to the modern one of protons and neutrons was not instantaneous. It took several years for the new paradigm to be generally accepted. What is less known is that the same year marks another revolution in our understanding of the fundamental interactions, which is closely related to the new nuclear model. It is the introduction of the concept of internal symmetries, a concept that introduced into theoretical physics a degree of abstraction not easy to grasp intuitively. The original idea is due to W. Heisenberg but, as we shall explain briefly later, the actual history is more complicated. We shall present the idea using our present understanding and we shall comment on the historical developments at the end.

We are used to the fact that particles may carry internal degrees of freedom. An electron at rest is described by two orthogonal states in the Hilbert space, the $|\uparrow\rangle$ with spin projection $+\frac{1}{2}$ and the $|\downarrow\rangle$ with spin projection $-\frac{1}{2}$. They satisfy the orthogonality relation $\langle\uparrow \mid \downarrow\rangle=0$. Normally, we could have invented two different names to describe these two states. Nevertheless, we talk about one electron and the reason why this is correct is that there exist transformations which leave the equations of motion invariant and transform one state to the other. Indeed, if the spin projections are taken along the $z$-axis, a rotation of $180^{\circ}$ around any axis in the $x-y$ plane interchanges $|\uparrow\rangle$ and $|\downarrow\rangle$.

物理代写|粒子物理代写Particle Physics代考|Fermi and the Weak Interactions

Already in 1926 , before the introduction of the Schrödinger wave equation, Fermi had published two papers with the statistical rules which established Fermi quantum statistics and gave fermions their name. In 1933, he came back with one of the most influential papers in particle physics in which he proposed a field theory model for the $\beta$-decay of neutrons. Even today, when this theory has been superseded by the Standard Model of electroweak interactions, which we shall develop in this book, Fermi’s theory is still used as a good low energy approximation.

This paper contains many revolutionary ideas. Fermi was one of the first physicists who believed in the physical existence of the neutrino. Contrary to Heisenberg, in the Bohr-Pauli controversy Fermi sided clearly with the second. But he went further and broke completely with the prevailing philosophy, according to which particles that come out from a nucleus ought to be present inside it. ${ }^{22}$ In his paper he formulated the full quantum field theory for fermion fields and introduced the formalism of creation and annihilation operators, the analogue of the ones we used for the electromagnetic field in equations (2.23)and (2.24). It was the first time that quantised fermion fields appeared in particle physics. We shall give a full account of this formalism in Chapter 10 where we shall also indicate the novel features that Fermi introduced in order to incorporate Fermi statistics. The paper appeared at the beginning of 1934 in Italian ${ }^{23}$ under the title Tentativo di una teoria della emissione di raggi $\beta$.

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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物理代写|粒子物理代写Particle Physics代考|B-beta-Decay and the Neutrino

The study of beta-decay played a very important role in the development of the entire field of subatomic physics, so we shall briefly review the main steps. We start with a simple kinematical analysis. Consider a particle $A$ which decays into two particles: $A\left(p_A\right) \rightarrow B\left(p_B\right)+C\left(p_C\right)$. The conservation of energy and momentum applied in the rest frame of the decaying particle implies
$$M_A=p_B^0+p_C^0, \quad 0=\boldsymbol{p}_B+\boldsymbol{p}_C$$
and gives
$$p_B^0=\frac{M_A^2+M_B^2-M_C^2}{2 M_A}, \quad p_C^0=\frac{M_A^2+M_C^2-M_B^2}{2 M_A}$$
i.e. the energy of each one of the final particles is fixed. In nuclear decays, both the electron mass $m_e \equiv M_C \simeq 0.5 \mathrm{MeV}$, and the mass difference $M_A-M_B$ between the initial and the final nuclei, typically of order a few $\mathrm{MeV}$, are negligible compared to the nuclear masses which are on the order of several $\mathrm{GeV}$. Therefore equation (6.3) tells us that, if $\beta$-decay is of the form given in (6.1), in the rest frame of the decaying nucleus the electrons are monoenergetic with energy given by
$$E_e \equiv p_C^0 \simeq \Delta M=M_A-M_B$$
The experiments performed up to 1914 did not show any appreciable contradiction with this result. We know today that this was due to lack of sufficient accuracy in the determination of the electron energy. The latter was estimated by the penetration length of the emitted electrons in various materials and the results were only approximate. The best measurements were performed in the chemistry department of the University of Berlin by O. Hahn, a former assistant of Rutherford, and L. Meitner.

物理代写|粒子物理代写Particle Physics代考|1932: The First Table of Elementary Particles

Table $6.1$ contains all the particles considered to be elementary in 1932 . We saw previously that the year is not chosen randomly. It is the year of the discovery of the neutron and the formulation of the correct nuclear model. Our basic ideas on the structure of matter date from that year. The table is separated into two parts. The upper part contains two doublets, the proton-neutron and the electron-neutrino which, as we shall explain shortly, can be considered as the constituents of matter. We shall call them matter particles. The lower part contains only one entry, the photon, which is the quantum manifestation of the electromagnetic field. We shall call it the quantum of radiation.
Let us have a first cursory look at each entry separately.

• The protons and the neutrons form the nuclei. We shall call them collectively nucleons. ${ }^5$ We shall see in one of the following sections that this is more than just a common name. They both have spin one-half and obey the Pauli exclusion principle.

Compared to the other particles in Table $6.1$, they appear to be heavy, with masses of order $1 \mathrm{GeV}$ and this fact explains why the mass of the atoms is contained almost entirely in the nuclei.

• The electron is the oldest-known elementary particle. It has a long history but its discovery is attributed to J.J. Thomson who, in 1897 , established that cathode rays consist of corpuscles carrying negative electric charge. Furthermore, he measured the ratio of charge over mass, $e / m$, quite accurately and determined that these properties are independent of the chemical composition of the cathode. In 1900, $\mathrm{H}$. Becquerel proved that the same particles are emitted in $\beta$-decay. The ratio $e / m$ was measured more accurately by R. Millikan in 1909 . The electrons also have spin one-half and obey the Pauli exclusion principle.
• We have already presented the story of the neutrino. ${ }^6$ In Table $6.1$ we see a question mark in the entry of its mass. It shows our ignorance concerning its precise value. We shall come back in more detail to the problem of the neutrino mass in Chapter 20.
• At the end of the matter particles in the table we see a line under the title “anti-particles”. When Dirac proposed his equation in 1928 , it was meant to be a relativistic wave equation for the electron. It was, however, soon realised by Dirac that the same equation admits solutions describing a positively charged particle with the same mass and spin as the electron. We shall introduce and study the Dirac equation in Chapter 7. Dirac thought for a while to identify this new solution with the proton and he published a paper along these lines in 1930, but he soon understood that it is in fact a new particle. In 1931, he predicted the existence of such an anti-electron, which we call the positron ${ }^7$ and the following year C.D. Anderson detected this particle ${ }^8$ in the cosmic rays using a cloud chamber, which was invented by C.T. Wilson two decades earlier. ${ }^9$ This successful prediction was a triumph of the Dirac theory. Concerning the antiparticle of the neutrino, we saw previously that, by convention whose origin will be clear later, we called anti-neutrino the particle conjectured by Pauli. So, in 1932 this was the only one that people knew. The existence of what we call today the “neutrino” was established in the same indirect way in 1934 when F. and I. Joliot-Curie discovered a reaction that looked like $\beta$-decay but the emitted particle was a positron rather than an electron. In the notation of nuclear physics $(A, Z)$ changes into $(A, Z-1)$ according to
$$(A, Z) \rightarrow(A, Z-1)+e^{+}+\nu$$

物理代写|粒子物理代写Particle Physics代考|B-beta\$-Decay and the Neutrino 学分析开始。考虑一个粒子$A$衰变为两个粒子:$A\left(p_A\right) \rightarrow B\left(p_B\right)+C\left(p_C\right)$. 在亯变粒子的静止框架中应用的能量和动量守恒 意味着 $$M_A=p_B^0+p_C^0, \quad 0=\boldsymbol{p}_B+\boldsymbol{p}_C$$ 并给出 $$p_B^0=\frac{M_A^2+M_B^2-M_C^2}{2 M_A}, \quad p_C^0=\frac{M_A^2+M_C^2-M_B^2}{2 M_A}$$ 即每个最終粒子的能量是固定的。在核言变中，电子质量$m_e \equiv M_C \simeq 0.5 \mathrm{MeV}$, 和质量差$M_A-M_B$在初始核和最终核之 间，通常为几个$\mathrm{MeV}$，与核质量相比可以忽略不计$\mathrm{GeV}$. 因此等式 (6.3) 告诉我们，如果$\beta$-哏变具有 (6.1) 中给出的形式，在 㚆变核的静止框架中，电子是单能的，能量由下式给出 $$E_e \equiv p_C^0 \simeq \Delta M=M_A-M_B$$ 直到 1914 年进行的实验并没有显示出与这个结果有任何明显的矛盾。我们今天知道这是由于在确定电子能量方面缺乏足够的淮确 性。后者是通过发射电子在各种材料中的穿透长度来估计的，结果只是近似值。最好的测量是由卢瑟福的前助理 O. Hahn 和 L. Meitner 在柏林大学的化学系进行的。 物理代写|粒子物理代写Particle Physics代考|1932: The First Table of Elementary Particles 桌子6.1包含所有在 1932 年被认为是基本的粒子。我们之前看到，年份不是随机选择的。这是发现中子和制定正确核模型的一 年。我们关于物质结构的基本思想是从那一年开始的。该表分为两部分。上半部分包含两个双峰，质子-中子和电子-中微子，正 如我们稍后将解释的，它们可以被认为是物质的组成部分。我们称它们为物质粒子。下半部分只包含一个条目，光子，它是电磁场 的量子表现。我们称之为辐射量子。 让我们先粗略地分别看一下每个条目。 • 质子和中子形成原子核。我们将它们统称为核子。我们将在以下部分中看到，这不仅仅是一个通用名称。他们都自旋了一 半并且邅守泡利不相容原理。 与表中其他粒子相比6.1，它们看起来很重, 有大量的秩序$1 \mathrm{GeV}$这个事实解释了为什么原子的质量几乎完全包含在原子核中。 • 电子是已知最古老的基本粒子。它有着悠久的历史，但它的发现归功于 JJ Thomson，他在 1897 年确立了阴极射线由带有 负电荷的微粒组成。此外，他测量了电荷与质量的比率，$e / m$，非常准确地确定这些特性与阴极的化学成分无关。1900 年，$\mathrm{H}$. 贝古勒尔证明了相同的粒子在$\beta$-亳变。比例$e / m R$. Millikan 在 1909 年更准确地测量了。电子也有二分之一的自旋 并邅守泡利不相容原理。 • 我们已经介绍了中微子的故事。${ }^6$在表中6.1我们在其质量的条目中看到了一个问号。它显示了我们对其精确价值的无知。 我们将在第 20 章更详细地讨论中微子质量问题。 • 在表中物质粒子的末尾，我们看到标题为“反粒子”的一行。当狄拉克在 1928 年提出他的方程时，它本来是一个电子的相对 论波动方程。然而，狄拉克很快意识到，同样的方程允许描述一个与电子具有相同质量和自旋的带正电粒子的解。我们将在 第 7 章介绍和研究狄拉克方程。狄拉克想了一阵子想用质子来确定这个新解，并在 1930 年发表了一篇沿此思路的论文，但 他很快就明白它实际上是一个新粒子。1931年，他预言了这种反电子的存在，我们称之为正电子${ }^7$第二年 CD Anderson 检 测到了这个粒子${ }^8$在宇宙射线中使用云室，这是由 CT Wilson 二十年前发明的。${ }^9$这一成功的预则是狄拉克理论的胜利。关 于中微子的反粒子，我们之前已经看到，按照贯例，其起源将在后面清楚，我们称反中微子为泡利猜想的粒子。所以，在 1932 年，这是人们唯一知道的。我们今天所说的“中微子”的存在是在 1934 年以同样的间接方式确立的，当时$\mathrm{F}$. 和$\mathrm{I}$. Joliot-Curie 发现了一种看起来像$\beta$-毫变，但发射的粒子是正电子而不是电子。在核物理符号中$(A, Z)$变成$(A, Z-1)$Joliot-C $$(A, Z) \rightarrow(A, Z-1)+e^{+}+\nu$$ 物理代写|粒子物理代写Particle Physics代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。 微观经济学代写 微观经济学是主流经济学的一个分支，研究个人和企业在做出有关稀缺资源分配的决策时的行为以及这些个人和企业之间的相互作用。my-assignmentexpert™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在图论代写Graph Theory代写方面经验极为丰富，各种图论代写Graph Theory相关的作业也就用不着 说。 线性代数代写 线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。 博弈论代写 现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。 微积分代写 微积分，最初被称为无穷小微积分或 “无穷小的微积分”，是对连续变化的数学研究，就像几何学是对形状的研究，而代数是对算术运算的概括研究一样。 它有两个主要分支，微分和积分；微分涉及瞬时变化率和曲线的斜率，而积分涉及数量的累积，以及曲线下或曲线之间的面积。这两个分支通过微积分的基本定理相互联系，它们利用了无限序列和无限级数收敛到一个明确定义的极限的基本概念 。 计量经济学代写 什么是计量经济学？ 计量经济学是统计学和数学模型的定量应用，使用数据来发展理论或测试经济学中的现有假设，并根据历史数据预测未来趋势。它对现实世界的数据进行统计试验，然后将结果与被测试的理论进行比较和对比。 根据你是对测试现有理论感兴趣，还是对利用现有数据在这些观察的基础上提出新的假设感兴趣，计量经济学可以细分为两大类：理论和应用。那些经常从事这种实践的人通常被称为计量经济学家。 MATLAB代写 MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 Posted on Categories:Particle Physics, 物理代写, 粒子物理 物理代写|粒子物理代写Particle Physics代考|PHY408 Tensor calculus 如果你也在 怎样代写粒子物理Particle Physics PHY408这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。粒子物理Particle Physics或高能物理学是对构成物质和辐射的基本粒子和力量的研究。宇宙中的基本粒子在标准模型中被分为费米子（物质粒子）和玻色子（载力粒子）。费米子有三代，但普通物质只由第一代费米子构成。第一代包括形成质子和中子的上下夸克，以及电子和电子中微子。已知由玻色子介导的三种基本相互作用是电磁力、弱相互作用和强相互作用。 粒子物理Particle Physics夸克不能单独存在，而是形成强子。含有奇数夸克的强子被称为重子，含有偶数夸克的强子被称为介子。两个重子，质子和中子，构成了普通物质的大部分质量。介子是不稳定的，寿命最长的介子只持续了几百分之一微秒的时间。它们发生在由夸克组成的粒子之间的碰撞之后，例如宇宙射线中快速移动的质子和中子。介子也会在回旋加速器或其他粒子加速器中产生。 粒子物理Particle Physics代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。最高质量的粒子物理Particle Physics作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此粒子物理Particle Physics作业代写的价格不固定。通常在专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。 avatest™帮您通过考试 avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！ 在不断发展的过程中，avatest™如今已经成长为论文代写，留学生作业代写服务行业的翘楚和国际领先的教育集团。全体成员以诚信为圆心，以专业为半径，以贴心的服务时刻陪伴着您， 用专业的力量帮助国外学子取得学业上的成功。 •最快12小时交付 •200+ 英语母语导师 •70分以下全额退款 想知道您作业确定的价格吗? 免费下单以相关学科的专家能了解具体的要求之后在1-3个小时就提出价格。专家的 报价比上列的价格能便宜好几倍。 我们在物理Physical代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的物理Physical代写服务。我们的专家在粒子物理Particle Physics代写方面经验极为丰富，各种粒子物理Particle Physics相关的作业也就用不着说。 物理代写|粒子物理代写Particle Physics代考|Tensor calculus Tensor calculus. The only noticeable difference between the tensor calculus in the Lorentz group and the corresponding one in any real orthogonal group is the non-Euclidean metric in$\mathbb{M}^4$which distinguishes between upper and lower indices. So, a contravariant four-vector$V^\mu$with$\mu=0,1,2,3$, is defined as a set of four quantities transforming as: $$V^{\prime \mu}=\Lambda^\mu{ }\nu V^\nu$$ Similarly, a set of four quantities$U\mu$define a covariant 4-vector if they transform under a Lorentz transformation as $$U_\mu^{\prime}=U_\nu\left(\Lambda^{-1}\right)^\nu{ }\mu \equiv \Lambda\mu^\nu U_\nu, \Lambda_\mu^\nu=\eta_{\mu \lambda} \Lambda_\rho^\lambda \eta^{\rho \nu}$$ The metric$\eta_{\mu \nu}$and its inverse$\eta^{\mu \nu}$can be used to lower and raise indices, respectively. These definitions extend to general contravariant or covariant or mixed tensors with arbitrary numbers of upper (contravariant) and/or lower (covariant) indices. The$4^{n+m}$quantities$T_{\nu_1 \nu_2 \ldots \nu_m}^{\mu_1 \mu_2 \ldots \mu_n}$are the components of a mixed tensor with$n$contravariant and$m$covariant indices, iff under Lorentz transformations they transform according to $$T_{\nu_1 \nu_2 \ldots \nu_m}^{\mu_1 \mu_2 \ldots \mu_n}=\Lambda_{\alpha_1}^{\mu_1} \Lambda_{\alpha_2}^{\mu_2} \ldots \Lambda_{\alpha_n}^{\mu_n}\left(\Lambda^{-1}\right){\nu_1}^{\beta_1}\left(\Lambda^{-1}\right){\nu_2}^{\beta_2} \ldots\left(\Lambda^{-1}\right){\nu_m}^{\beta_m} T{\beta_1 \beta_2 \ldots \beta_m}^{\alpha_1 \alpha_2 \ldots \alpha_n}$$ A quantity which is invariant under Lorentz transformations is called a Lorentz scalar. A tensor without any index or with all its indices contracted is a Lorentz scalar. In analogy to tensors of the rotation group$O(3)$, Lorentz tensors decompose into irreducible ones transforming independently with the help of the operations of symmetrisation, anti symmetrisation and trace. 物理代写|粒子物理代写Particle Physics代考|The Lie algebra of the Lorentz group The Lie algebra of the Lorentz group. The proper Lorentz group contains the spatial rotations as a subgroup. Indeed, the matrices $$\Lambda=\left(\begin{array}{ll} 1 & \ & R \end{array}\right)$$ with the three-by-three matrices$R$being rotation matrices$R^T R=R R^T=1$, satisfy (5.131). The rotation matrix in the ”$x^1-x^2$plane” by angle$\omega_{12}=\theta$is $$R\left(\omega_{12}=\theta\right)=\left(\begin{array}{ccc} \cos \theta & \sin \theta & 0 \ -\sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{array}\right)$$ and the corresponding generator defined in general in (5.21) is $$\mathcal{J}^{12}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & \mathrm{i} & 0 \ 0 & -\mathrm{i} & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right)$$ Similarly, the generators of rotations in the 2-3 and 3-1 planes are $$\mathcal{J}^{23}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & -\mathrm{i} & 0 \end{array}\right) \quad \text { and } \mathcal{J}^{13}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & 0 & 0 \ 0 & -\mathrm{i} & 0 & 0 \end{array}\right)$$ The Lorentz boost with velocity$V$in the$x^1$-direction is given by $$t^{\prime}=\gamma(V)(t-V x), \quad x^{\prime}=\gamma(V)(x-V t), \quad y^{\prime}=y, \quad z^{\prime}=z$$ with $$\gamma(V)=\frac{1}{\sqrt{1-V^2}}$$ and similarly for boosts in the other two directions. From (5.143) we obtain for the generators of boosts $$\mathcal{J}^{01}=-\mathrm{i}\left(\begin{array}{llll} 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{02}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{03}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{array}\right)$$ 粒子物理代写 物理代写|粒子物理代写粒子物理代考|张量微积分 张量微积分。洛伦兹群中的张量微积分和任何实正交群中的相应张量微积分之间唯一明显的区别是$\mathbb{M}^4$中的非欧氏度规，它区分了上指标和下指标。因此，一个逆变四向量$V^\mu$和$\mu=0,1,2,3$被定义为一组四个量的变换为: $$V^{\prime \mu}=\Lambda^\mu{ }\nu V^\nu$$类似地，一组四个量$U\mu$定义一个协变四向量，如果它们在洛伦兹变换下变换为 $$U_\mu^{\prime}=U_\nu\left(\Lambda^{-1}\right)^\nu{ }\mu \equiv \Lambda\mu^\nu U_\nu, \Lambda_\mu^\nu=\eta_{\mu \lambda} \Lambda_\rho^\lambda \eta^{\rho \nu}$$ 度量$\eta_{\mu \nu}$和它的逆$\eta^{\mu \nu}$可以分别用来降低和提高指数。这些定义扩展到具有任意数量上(逆变)和/或下(协变)指标的一般逆变或协变或混合张量。$4^{n+m}$量$T_{\nu_1 \nu_2 \ldots \nu_m}^{\mu_1 \mu_2 \ldots \mu_n}$是具有$n$逆变指标和$m$协变指标的混合张量的分量，iff在洛伦兹变换下它们根据 $$T_{\nu_1 \nu_2 \ldots \nu_m}^{\mu_1 \mu_2 \ldots \mu_n}=\Lambda_{\alpha_1}^{\mu_1} \Lambda_{\alpha_2}^{\mu_2} \ldots \Lambda_{\alpha_n}^{\mu_n}\left(\Lambda^{-1}\right){\nu_1}^{\beta_1}\left(\Lambda^{-1}\right){\nu_2}^{\beta_2} \ldots\left(\Lambda^{-1}\right){\nu_m}^{\beta_m} T{\beta_1 \beta_2 \ldots \beta_m}^{\alpha_1 \alpha_2 \ldots \alpha_n}$$ 进行变换，在洛伦兹变换下不变的量称为洛伦兹标量。一个没有任何指标或所有指标收缩的张量是洛伦兹标量 类似于旋转群的张量$O(3)$，洛伦兹张量在对称、反对称和迹化操作的帮助下分解为独立变换的不可约张量 物理代写|粒子物理代写粒子物理学代考|洛伦兹群的李代数 洛伦兹群的李代数。真正的洛伦兹群包含空间旋转作为子群。确实，矩阵 $$\Lambda=\left(\begin{array}{ll} 1 & \ & R \end{array}\right)$$ ， 3 × 3矩阵$R$是旋转矩阵$R^T R=R R^T=1$，满足(5.131)。“$x^1-x^2$平面”中通过角度$\omega_{12}=\theta$的旋转矩阵为 $$R\left(\omega_{12}=\theta\right)=\left(\begin{array}{ccc} \cos \theta & \sin \theta & 0 \ -\sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{array}\right)$$ ，对应的生成器在(5.21)中一般定义为 $$\mathcal{J}^{12}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & \mathrm{i} & 0 \ 0 & -\mathrm{i} & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right)$$ 类似地，在2-3和3-1平面中的旋转生成器为 $$\mathcal{J}^{23}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & -\mathrm{i} & 0 \end{array}\right) \quad \text { and } \mathcal{J}^{13}=\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & \mathrm{i} \ 0 & 0 & 0 & 0 \ 0 & -\mathrm{i} & 0 & 0 \end{array}\right)$$$x^1$方向上速度$V$的洛伦兹升力由 $$t^{\prime}=\gamma(V)(t-V x), \quad x^{\prime}=\gamma(V)(x-V t), \quad y^{\prime}=y, \quad z^{\prime}=z$$ with $$\gamma(V)=\frac{1}{\sqrt{1-V^2}}$$ 给出，其他两个方向上的升力也类似。由(5.143)我们得到对于boost $$\mathcal{J}^{01}=-\mathrm{i}\left(\begin{array}{llll} 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{02}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right), \mathcal{J}^{03}=-\mathrm{i}\left(\begin{array}{llll} 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{array}\right)$$ 的生成器 物理代写|粒子物理代写Particle Physics代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。 微观经济学代写 微观经济学是主流经济学的一个分支，研究个人和企业在做出有关稀缺资源分配的决策时的行为以及这些个人和企业之间的相互作用。my-assignmentexpert™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在图论代写Graph Theory代写方面经验极为丰富，各种图论代写Graph Theory相关的作业也就用不着 说。 线性代数代写 线性代数是数学的一个分支，涉及线性方程，如：线性图，如：以及它们在向量空间和通过矩阵的表示。线性代数是几乎所有数学领域的核心。 博弈论代写 现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。 微积分代写 微积分，最初被称为无穷小微积分或 “无穷小的微积分”，是对连续变化的数学研究，就像几何学是对形状的研究，而代数是对算术运算的概括研究一样。 它有两个主要分支，微分和积分；微分涉及瞬时变化率和曲线的斜率，而积分涉及数量的累积，以及曲线下或曲线之间的面积。这两个分支通过微积分的基本定理相互联系，它们利用了无限序列和无限级数收敛到一个明确定义的极限的基本概念 。 计量经济学代写 什么是计量经济学？ 计量经济学是统计学和数学模型的定量应用，使用数据来发展理论或测试经济学中的现有假设，并根据历史数据预测未来趋势。它对现实世界的数据进行统计试验，然后将结果与被测试的理论进行比较和对比。 根据你是对测试现有理论感兴趣，还是对利用现有数据在这些观察的基础上提出新的假设感兴趣，计量经济学可以细分为两大类：理论和应用。那些经常从事这种实践的人通常被称为计量经济学家。 MATLAB代写 MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 Posted on Categories:Particle Physics, 物理代写, 粒子物理 物理代写|粒子物理代写Particle Physics代考|PHYS125 The group SU(3) 如果你也在 怎样代写粒子物理Particle Physics PHYS125这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。粒子物理Particle Physics或高能物理学是对构成物质和辐射的基本粒子和力量的研究。宇宙中的基本粒子在标准模型中被分为费米子（物质粒子）和玻色子（载力粒子）。费米子有三代，但普通物质只由第一代费米子构成。第一代包括形成质子和中子的上下夸克，以及电子和电子中微子。已知由玻色子介导的三种基本相互作用是电磁力、弱相互作用和强相互作用。 粒子物理Particle Physics夸克不能单独存在，而是形成强子。含有奇数夸克的强子被称为重子，含有偶数夸克的强子被称为介子。两个重子，质子和中子，构成了普通物质的大部分质量。介子是不稳定的，寿命最长的介子只持续了几百分之一微秒的时间。它们发生在由夸克组成的粒子之间的碰撞之后，例如宇宙射线中快速移动的质子和中子。介子也会在回旋加速器或其他粒子加速器中产生。 粒子物理Particle Physics代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。最高质量的粒子物理Particle Physics作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此粒子物理Particle Physics作业代写的价格不固定。通常在专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。 avatest™帮您通过考试 avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！ 在不断发展的过程中，avatest™如今已经成长为论文代写，留学生作业代写服务行业的翘楚和国际领先的教育集团。全体成员以诚信为圆心，以专业为半径，以贴心的服务时刻陪伴着您， 用专业的力量帮助国外学子取得学业上的成功。 •最快12小时交付 •200+ 英语母语导师 •70分以下全额退款 想知道您作业确定的价格吗? 免费下单以相关学科的专家能了解具体的要求之后在1-3个小时就提出价格。专家的 报价比上列的价格能便宜好几倍。 我们在物理Physical代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的物理Physical代写服务。我们的专家在粒子物理Particle Physics代写方面经验极为丰富，各种粒子物理Particle Physics相关的作业也就用不着说。 物理代写|粒子物理代写Particle Physics代考|The group SU(3) In this chapter we shall describe the main properties of the group$S U(3)$and its representations. As will become clear in the following chapters,$S U(3)$is related to a symmetry of fundamental importance in particle physics. Here we shall present its mathematical structure.$S U(3)$is the group which is isomorphic to that of unitary$3 \times 3$matrices of determinant equal to$+1$. Thus, the general element$u$satisfies the conditions $$u^{\dagger} u=\mathbf{1}=u u^{\dagger} \text { and } \operatorname{det} u=1$$ and, consequently, is characterised by eight real parameters. 物理代写|粒子物理代写Particle Physics代考|The representations of$\mathbf SU(3)

The representations of $\mathbf{S U ( 3 )}$. To construct the irreducible representations of $S U(3)$ and the corresponding Clebsch-Gordan decomposition rules, we shall follow the tensor method used for $S U(2)$. Modulo a few special features of $S U(2)$ and the combinatoric complications as we move on to higher groups and representations, the method is applicable to all members of the unitary series and even beyond.

• The trivial (singlet) representation 1. As usual, this is the representation in which all elements of the group are represented by the number 1 .
• The defining (3) representation. Three complex quantities $\psi^i, i=1,2,3$, are said to be the components of a triplet (3) of $S U(3)$, if they transform according to
$$\psi^{i^{\prime}}=u_j^i \psi^i \quad\left(\psi^{\prime}=u \psi\right) \forall u \in S U(3)$$
• The conjugate $\overline{\mathbf{3}}$ of the defining representation. Three complex quantities $\chi_i, i=1,2,3$, are defined to be the components of an anti-triplet $\overline{\mathbf{3}}$ of $S U(3)$, if they transform according to
$$\chi_i^{\prime}=u_i^{\dagger j} \chi_j \quad\left(\chi^{\prime}=\chi u^{\dagger}\right) \forall u \in S U(3)$$
An immediate consequence of this definition is that, if $\psi^i$ is a triplet, the complex conjugates $\psi^{i^*}$ form an anti-triplet, since then, $\psi^{\dagger}$, whose components are exactly the $\psi^{i *}$, indeed transform according to ${\psi^{\prime}}^{\prime}=\psi^{\dagger} u^{\dagger}$. Thus, the consistent notation of the components of $\psi^{\dagger}$ is $\psi_i^{\dagger}$ with one lower index.

Also, notice that given two triplets $\psi$ and $\chi$, the quantities $\psi^{\dagger} \chi=\psi^{i *} \chi^i$ and its Hermitian conjugate $\chi^{\dagger} \psi$ are invariant (singlets) under $S U(3)$.

物理代写|粒子物理代写Particle Physics代考|The group SU(3)

$S U(3)$ 是与酉同构的群 $3 \times 3$ 行列式矩阵等于 $+1$. 因此，一般元龶 $u$ 满足条件
$$u^{\dagger} u=\mathbf{1}=u u^{\dagger} \text { and } \operatorname{det} u=1$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。